A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces

et al. 2021

Preprint

Self Cite

We consider the numerical algorithm for the two-dimensional time-harmonic elastic wave scattering by unbounded rough surfaces with Dirichlet boundary condition. A Nyström method is proposed for the scattering problem based on the integral equation method. Convergence of the Nyström method is established with convergence rate depending on the smoothness of the rough surfaces. In doing so, a crucial role is played by analyzing the singularities of the kernels of the relevant boundary integral operators. Numerical experiments are presented to demonstrate the effectiveness of the method.

Section: Introductionmentioning

confidence: 99%

The Nyström method for elastic wave scattering by unbounded rough surfaces

Liu²,

et al. 2021

Preprint

Self Cite

“…Many inversion algorithms have been proposed in the literature if the rough surface is scattered by an incident acoustic field. For instance, a nonlinear integral equation method was proposed in [5] for recovering an impenetrable rough surface with a Dirichlet boundary condition, and a direct sampling method was proposed in [6] for imaging an impenetrable surface or an interface in dielectric media by taking near-field Cauchy data. If the rough surface is assumed to be a small and smooth deformation of a plane, a transformed field expansion method was proposed in [7] for recovering the surface with a Dirichlet, impedance or transmission condition, and a factorization method was proposed in [8] for recovering a Dirichlet rough surface under the assumption κf + < √ 2, where κ > 0 stands for the wavenumber and f + stands for the height of the rough surface.…”

Section: Introductionmentioning

confidence: 99%

A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces

Liu

et al. 2019

Inverse Problems

This paper is concerned with the inverse time-harmonic elastic scattering problem of recovering unbounded rough surfaces in two dimensions. We assume that elastic plane waves with different directions are incident onto a rigid rough surface in a half plane. The elastic scattered field is measured on a horizontal straight line segment within a finite distance above the rough surface. A direct imaging algorithm is proposed to recover the unbounded rough surface from the scattered near-field data, which involves only inner products between the data. Numerical experiments are presented to show that the inversion scheme is not only efficient but also accurate and robust with respect to noise.

“…We point out that many reconstruction algorithms have also been developed for reconstructing non-locally rough surfaces from the scattered near-field data (see, e.g. [4,5,9,10,16,21,22,31]).…”

Section: Introductionmentioning

confidence: 99%

Imaging of locally rough surfaces from intensity-only far-field or near-field data

2017

Inverse Problems

This paper is concerned with a nonlinear imaging problem, which aims to reconstruct a locally perturbed, perfectly reflecting, infinite plane from intensity-only (or phaseless) far-field or near-field data. A recursive Newton iteration algorithm in frequencies is developed to reconstruct the locally rough surface from multi-frequency intensity-only far-field or near-field data, where the fast integral equation solver developed in [39] is used to solve the direct scattering problem in each iteration. For the case with far-field data, a main feature of our work is that the incident field is taken as a superposition of two plane waves with different directions rather than one plane wave, so the location and shape of the local perturbation of the infinite plane can be reconstructed simultaneously from intensity-only far-field data with multiple wave numbers. This is different from previous work on inverse scattering from phaseless far-field data, where only the shape reconstruction was considered due to the translation invariance property of the phaseless far-field pattern corresponding to one plane wave as the incident field. Finally, numerical examples are carried out to demonstrate that our reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.